A034807 → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(-n - n x + n^{2} + n^{3} x\right)\,P_{n}(x) = \left(-4\,x^{2} - n - 5\,n x - 2\,n x^{2} + n^{2} + 4\,n^{2} x + 4\,n^{2} x^{2} + n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x^{2} - 8\,x^{3} - n x - 7\,n x^{2} - 12\,n x^{3} - n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	2				2
1	1				1
2	1	4			5
3	1	6			7
4	1	8	6		15
5	1	10	15		26
6	1	12	27	8	48
7	1	14	42	28	85

All 15 rows

Technical details

Node id: n-d15b0c03f5b112b4 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-11372483dbfbeb3e

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 2;
  1: 1;
  2: 1,  4;
  3: 1,  6;
  4: 1,  8,  6;
  5: 1, 10, 15;
  6: 1, 12, 27,   8;
  7: 1, 14, 42,  28;
  8: 1, 16, 60,  64, 10;
  9: 1, 18, 81, 120, 45;

%F Formula

The row polynomials satisfy (n^2-n+(n^3-n)*x)*P(n,x) = (n^2-n+(n^3+4*n^2-5*n)*x+(2*n^3+4*n^2-2*n-4)*x^2)*P(n-1,x) - (n*x+(n^2+7*n+2)*x^2+(4*n^2+12*n+8)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 1.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - n + (n^3 + 4*n^2 - 5*n)*x + (2*n^3 + 4*n^2 - 2*n - 4)*x^2)*P[n - 1, x] - (n*x + (n^2 + 7*n + 2)*x^2 + (4*n^2 + 12*n + 8)*x^3)*D[P[n - 1, x], x])/(n^2 - n + (n^3 - n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 - n + (n^3 + 4*n^2 - 5*n)*x + (2*n^3 + 4*n^2 - 2*n - 4)*x^2)*v[i - 1] - (n*x + (n^2 + 7*n + 2)*x^2 + (4*n^2 + 12*n + 8)*x^3)*deriv(v[i - 1], x))/(n^2 - n + (n^3 - n)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

reversed rows → θ+1

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.